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Macroeconomics Exam Review 244

# Macroeconomics Exam Review 244 - Solutions for Foundations...

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5.45 For each 𝑗 = 1 , 2 , . . . , 𝑚 , either 𝑔 𝑗 ( x ) < 0 which implies that 𝜆 𝑗 = 0 and therefore 𝜆 𝑗 𝐷𝑔 𝑗 [ x ]( x x ) = 0 or 𝑔 𝑗 ( x ) = 0 Since 𝑔 𝑗 is quasiconvex and 𝑔 𝑗 ( x ) 0 = 𝑔 ( x ), Exercise 4.73 implies that 𝐷𝑔 𝑗 [ x ]( x x ) 0. Since 𝜆 𝑗 0, this implies that 𝜆 𝑗 𝐷𝑔 𝑗 [ x ]( x x ) 0. We have shown that for every 𝑗 , 𝜆 𝑗 𝐷𝑔 𝑗 [ x ]( x x ) 0. The first-order condition implies that 𝐷𝑓 [ x ]( x x ) = 𝑗 𝜆 𝑗 𝐷𝑔 𝑗 [ x ]( x x ) 0 If 𝑓 ( x ) 𝜆 𝑗 𝑔 𝑗 ( x ) x 0 ( 𝑓 ( x ) 𝜆 𝑗 𝑔 𝑗 ( x ) ) 𝑇 x = 0 The first-order conditions imply that for every x 𝐺 , x 0 and ( 𝑓 ( x ) 𝜆 𝑗 𝑔 𝑗 ( x ) ) 𝑇 x 0 and therefore ( 𝑓 ( x ) 𝜆 𝑗 𝑔 𝑗 ( x ) ) 𝑇 ( x x ) 0 or 𝑓 ( x ) 𝑇 ( x x ) 𝜆 𝑗 𝑔 𝑗 ( x ) 𝑇 ( x x ) 0 5.46 Assuming 𝑥 𝑑 = 𝑥 𝑑 = 0, the constraints become
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